Inversion (discrete mathematics)

Let $\backslash pi$ be a permutation.

There is an inversion of $\backslash pi$ between $i$ and $j$ if and $\backslash pi(i)\; >\; \backslash pi(j)$. The inversion is indicated by an ordered pair containing either the places $(i,\; j)${{sfn|Aigner|2007|pp=27}}{{sfn|Comtet|1974|pp=237}} or the elements $\backslash bigl(\backslash pi(i),\; \backslash pi(j)\backslash bigr)$.{{sfn|Knuth|1973|pp=11}}{{sfn|Pemmaraju|Skiena|2003|pp=69}}{{sfn|Vitter|Flajolet|1990|pp=459}}

The inversion set is the set of all inversions. A permutation's inversion set using place-based notation is the same as the inverse permutation's inversion set using element-based notation with the two components of each ordered pair exchanged. Likewise, a permutation's inversion set using element-based notation is the same as the inverse permutation's inversion set using place-based notation with the two components of each ordered pair exchanged.{{sfn|Gratzer|2016|pp=221}}

Inversions are usually defined for permutations, but may also be defined for sequences:
Let $S$ be a sequence (or multiset permutation{{sfn|Bóna|2012|pp=57}}). If and $S(i)\; >\; S(j)$, either the pair of places $(i,\; j)${{sfn|Bóna|2012|pp=57}}{{sfn|Cormen|Leiserson|Rivest|Stein|2001|pp=39}} or the pair of elements $\backslash bigl(S(i),\; S(j)\backslash bigr)${{sfn|Barth|Mutzel|2004|pp=183}} is called an inversion of $S$.

For sequences, inversions according to the element-based definition are not unique, because different pairs of places may have the same pair of values.

The inversion number $\backslash mathtt\{inv\}(X)${{sfn|Mannila|1985}} of a sequence $X=\backslash langle\; x\_1,\backslash dots,x\_n\backslash rangle$, is the cardinality of the inversion set. It is a common measure of sortedness (sometimes called presortedness) of a permutation{{sfn|Vitter|Flajolet|1990|pp=459}} or sequence.{{sfn|Barth|Mutzel|2004|pp=183}} The inversion number is between 0 and $\backslash frac\{n(n-1)\}2$ inclusive. A permutation and its inverse have the same inversion number.

For example $\backslash mathtt\{inv\}(\backslash langle1,2,\backslash dots,\; n\backslash rangle)=0$ since the sequence is ordered. Also, when $n\; =\; 2m$ is even, $\backslash mathtt\{inv\}(\backslash langle\; m+1,m+2,\backslash dots,2m,1,2,\backslash dots,\; m\backslash rangle)=m^2$ (because each pair is an inversion). This last example shows that a set that is intuitively "nearly sorted" can still have a quadratic number of inversions.

The inversion number is the number of crossings in the arrow diagram of the permutation,{{sfn|Gratzer|2016|pp=221}} the permutation's Kendall tau distance from the identity permutation, and the sum of each of the inversion related vectors defined below.

Other measures of sortedness include the minimum number of elements that can be deleted from the sequence to yield a fully sorted sequence, the number and lengths of sorted "runs" within the sequence, the Spearman footrule (sum of distances of each element from its sorted position), and the smallest number of exchanges needed to sort the sequence.{{sfn|Mahmoud|2000|pp=284}} Standard comparison sorting algorithms can be adapted to compute the inversion number in time {{math|O(n log n)}}.{{sfn|Kleinberg|Tardos|2005|pp=225}}

Three similar vectors are in use that condense the inversions of a permutation into a vector that uniquely determines it. They are often called inversion vector or Lehmer code. (A list of sources is found here.)

This article uses the term inversion vector ($v$) like Wolfram.Weisstein, Eric W. [http://mathworld.wolfram.com/InversionVector.html "Inversion Vector"] From MathWorld--A Wolfram Web Resource The remaining two vectors are sometimes called left and right inversion vector, but to avoid confusion with the inversion vector this article calls them left inversion count ($l$) and right inversion count ($r$). Interpreted as a factorial number the left inversion count gives the permutations reverse colexicographic,Reverse colex order of finitary permutations {{OEIS|A055089}} and the right inversion count gives the lexicographic index.

File:Inversion example; Rothe 1.svg

Inversion vector $v$:
With the element-based definition $v(i)$ is the number of inversions whose smaller (right) component is $i$.{{sfn|Knuth|1973|pp=11}}

:$v(i)$ is the number of elements in $\backslash pi$ greater than $i$ before $i$.

:

Left inversion count $l$:
With the place-based definition $l(i)$ is the number of inversions whose bigger (right) component is $i$.

:$l(i)$ is the number of elements in $\backslash pi$ greater than $\backslash pi(i)$ before $\backslash pi(i)$.

:

Right inversion count $r$, often called Lehmer code:
With the place-based definition $r(i)$ is the number of inversions whose smaller (left) component is $i$.

:$r(i)$ is the number of elements in $\backslash pi$ smaller than $\backslash pi(i)$ after $\backslash pi(i)$.

:

Both $v$ and $r$ can be found with the help of a Rothe diagram, which is a permutation matrix with the 1s represented by dots, and an inversion (often represented by a cross) in every position that has a dot to the right and below it. $r(i)$ is the sum of inversions in row $i$ of the Rothe diagram, while $v(i)$ is the sum of inversions in column $i$. The permutation matrix of the inverse is the transpose, therefore $v$ of a permutation is $r$ of its inverse, and vice versa.

File:2-element subsets of 4 elements; array, hexagonal.svg

The following sortable table shows the 24 permutations of four elements (in the $\backslash pi$ column) with their place-based inversion sets (in the p-b column), inversion related vectors (in the $v$, $l$, and $r$ columns), and inversion numbers (in the # column). (The columns with smaller print and no heading are reflections of the columns next to them, and can be used to sort them in colexicographic order.)

It can be seen that $v$ and $l$ always have the same digits, and that $l$ and $r$ are both related to the place-based inversion set. The nontrivial elements of $l$ are the sums of the descending diagonals of the shown triangle, and those of $r$ are the sums of the ascending diagonals. (Pairs in descending diagonals have the right components 2, 3, 4 in common, while pairs in ascending diagonals have the left components 1, 2, 3 in common.)

The default order of the table is reverse colex order by $\backslash pi$, which is the same as colex order by $l$. Lex order by $\backslash pi$ is the same as lex order by $r$.

{{4-el perm inversions}}

File:Symmetric group 4; permutohedron 3D; numbers.svg S₄]]

The set of permutations on n items can be given the structure of a partial order, called the weak order of permutations, which forms a lattice.

The Hasse diagram of the inversion sets ordered by the subset relation forms the skeleton of a permutohedron.

If a permutation is assigned to each inversion set using the place-based definition, the resulting order of permutations is that of the permutohedron, where an edge corresponds to the swapping of two elements with consecutive values. This is the weak order of permutations. The identity is its minimum, and the permutation formed by reversing the identity is its maximum.

If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph, where an edge corresponds to the swapping of two elements on consecutive places. This Cayley graph of the symmetric group is similar to its permutohedron, but with each permutation replaced by its inverse.

{{wikiversity|Inversion (discrete mathematics)}}

• Factorial number system
• Permutation graph
• Transpositions, simple transpositions, inversions and sorting
• Damerau–Levenshtein distance
• Parity of a permutation

Sequences in the OEIS:

• [https://oeis.org/wiki/Index_to_OEIS:_Section_Fa#facbase Sequences related to factorial base representation]
• Factorial numbers: {{OEIS link|A007623}} and {{OEIS link|A108731}}
• Inversion numbers: {{OEIS link|A034968}}
• Inversion sets of finite permutations interpreted as binary numbers: {{OEIS link|A211362}}   (related permutation: {{OEIS link|A211363}})
• Finite permutations that have only 0s and 1s in their inversion vectors: {{OEIS link|A059590}}   (their inversion sets: {{OEIS link|A211364}})
• Number of permutations of n elements with k inversions; Mahonian numbers: {{OEIS link|A008302}}   (their row maxima; Kendall-Mann numbers: {{OEIS link|A000140}})
• Number of connected labeled graphs with n edges and n nodes: {{OEIS link|A057500}}

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